Optimal Codes for Correcting a Single (Wrap-Around) Burst of Erasures

In 2007, Martinian and Trott presented codes for correcting a burst of erasures with a minimum decoding delay. Their construction employs $[n,k]$ codes that can correct any burst of erasures (including wrap-around bursts) of length $n-k$ . They raised the question if such $[n,k]$ codes exist for all integers $k$ and $n$ with $1\leq k\leq n$ and all fields (in particular, for the binary field). In this correspondence, we answer this question affirmatively by giving two recursive constructions and a direct one.